Abstract
In this paper, we consider a more general form of vector variational-like inequalities for multivalued maps and prove some results on the existence of solutions of our new class of vector variational-like inequalities in the setting of topological vector spaces. Several special cases were also discussed.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Q. H. Ansari, “On generalized vector variational-like inequalities”. Ann. Sci. Mathem. Québec, Vol. 19, 1995, pp. 131–137.
Q. H. Ansari, “Extended generelized vector variational-like inequalities for nonmonotone multivalued maps”. Ann. Sci. Mathem. Québec, Vol. 21, 1997, pp. 1–11.
Q. H. Ansari, “A note on generalized vector variational–like inequalities”. Optimization, Gordon &Breach, Vol. 41, 1997, pp. 197–205.
Q. H. Ansari and A. H. Siddiqi, “A generalized vector variational-like inequality and optimization over an efficient set”. In “Functional analysis with current applications in science, technology and industry”, M. Brokate and A. H. Siddiqi Eds., Pitman Research Notes in Mathematics Series, No. 377, Longman, Essex, 1998, pp. 177–191.
F. E. Browder, “The fixed point theory of multivalued map**s in topological vector spaces”. Mathem. Ann., Vol. 177, 1968, pp. 283–301.
G.-Y. Chen, “Existence of solutions for a vector variational inequality: an extension of Hartmann-Stampacchia theorem”. Jou. Optimiz. Theory Appls., Vol. 74, 1992, pp. 445–456.
G.-Y. Chen and G. M. Cheng, “Vector variational inequality and vector optimization”. In “Lecture Notes in Econ. and Mathem. Systems, No. 285, Springer-Verlag, 1987, pp. 408–416.
G.-Y. Chen and B. D. Craven, “Approximate dual and approximate vector variational inequality for multiobjective optimization”. Jou. Austral. Mathem. Soc., Vol. 47, Series A, 1989, pp. 418–423.
G.-Y. Chen and B. D. Craven, “A vector variational inequality and optimization over an efficient set”. ZOR-Meth. Model. Operations Research, Vol. 3, 1990, pp. 1–12.
G.-Y. Chen and X. Q. Yang, “The vector complementarity problem and its equivalences with the weak minimal element in ordered spaces”. Jou. Mathem. Analysis and Appls., Vol. 153, 1990, pp. 136–158.
M. S. R. Chowdhury and K. K. Tan, “Generalized variational inequalities for quasi-monotone operators and applications”. Bull. Polish Acad. Sci. Mathem., Vol. 45, 1997, pp. 25–54.
J. B. Conway, “A course in functional analysis”. Springer-Verlag, New York, 1990.
A. Daniilidis and N. Hadjisavvas, “Existence theorems for vector variational inequalities”. Bull. Austral. Mathem. Soc., Vol. 54, 1996, pp. 473–481.
K. Fan, “A generalization of Tychonoff’s fixed point theorem”. Mathem. Ann., Vol. 142, 1961, pp. 305–310.
J. Fu, “Simultaneous vector variational inequalities and vector implicit complementarity problem”. Jou. Optimiz. Theory Appls., Vol. 93, 1997, pp. 141–151.
F. Giannessi, “Theorems of alternative, quadratic programs and complementarity problems”. In “Variational inequalities and complementarity problems”, R.W. Cottle, F. Giannessi and J.-L. Lions Eds., John Wiley and Sons, Chichester, 1980, pp. 151–186.
H. Kneser, “Sur un theoreme fondamantal de la theorie des jeux”. C. R. Acad. Sci. Paris, Vol. 234, 1952, pp. 2418–2420.
I. V. Konnov and J. C. Yao, “On the generalized vector variational inequality problem”. Jou. Mathem. Analysis and Appls., Vol. 206, 1997, pp. 42–58.
T. C. Lai and J. C. Yao, “Existence results for VVIP”. Appl. Mathem. Lett., Vol. 9, 1996, pp. 17–19.
G. M. Lee, D. S. Kim and B. S. Lee, “Generalized vector variational inequality”. Appl. Mathem. Lett., Vol. 9, 1996, pp. 39–42.
G. M. Lee, D. S. Kim, B. S. Lee and S. J. Cho, “Generalized vector variational inequality and fuzzy extension”. Appl. Mathem. Lett., Vol. 6, 1993, pp. 47–51.
G. M. Lee, D. S. Kim, B. S. Lee and N. D. Yen, “Vector variational inequality as a tool for studying vector optimization problems”. Nonlinear Analysis, Theory, Meth. Appls., Vol. 34, 1998, pp. 745–765.
G. M. Lee and S. Kum, “On implicit vector variational inequalities”. Jou. Optimiz. Theory Appls. (to appear).
B. S. Lee, G. M. Lee and D. S. Kim, “Generalized vector-valued variational inequalities and fuzzy extension”. Jou. Korean Mathem. Soc., Vol. 33, 1996, pp. 609–624.
B. S. Lee, G. M. Lee and D. S. Kim, “Generalized vector variational-like inequalities in locally convex Hausdorff topological vector spaces”. Indian Jou. Pure Appl. Mathem., Vol. 28, 1997, pp. 33–41.
L. J. Lin, “Pre-vector variational inequalities”. Bull. Austral. Mathem. Soc., Vol. 53, 1996, pp. 63–70.
K. L. Lin, D. P. Yang and J. C. Yao, “Generalized vector variational inequalities”. Jou. Optimiz. Theory Appls., Vol. 92, 1997, pp. 117–125.
J. Parida and A. Sen, “A variational-like inequality for multifunctions with applications” Jou. Mathem. Analysis Appls., Vol. 124, 1987, pp. 73–81.
A. H. Siddiqi, Q. H. Ansari, and A. Khaliq, “On vector variational inequalities”. Jou. Optimiz. Theory Appls., Vol. 84, 1995, pp. 171–180.
A. H. Siddiqi, Q. H. Ansari and R. Ahmad, “On vector variational-like inequalities”. Indian Jou. Pure Appl. Mathem., Vol. 28, 1997, pp. 1009–1016.
A. H. Siddiqi, Q. H. Ansari and M. F. Khan, “Variational-like inequalities for multivalued maps”. Indian Jou. Pure Appl. Mathem., Vol. 3, No. 2, 1999, pp. 161–166.
X. Q. Yang, “Generalized convex functions and vector variational inequalities”. Jou. Optimiz. Theory Appls., Vol. 79, 1993, pp. 563–580.
X. Q. Yang, “Vector complementarity and minimal element problems”. Jou. Optimiz. Theory Appls., 77, 1993, pp. 483–495.
X. Q. Yang, “Vector variational inequality and its duality”. Nonlinear Analysis, Theory, Meth. Appls., Vol. 21, 1993, pp. 869–877.
X. Q. Yang and C. J. Goh, “On vector variational inequalities: application to vector equilibria”. Jou. Optimiz. Theory Appls., Vol. 95, 1997, pp. 431–443.
X. Q. Yang, “Vector variational inequality and vector pseudolinear optimization”. Jou. Optimiz. Theory Appls., Vol. 95, 1997, pp. 729–734.
J. C. Yao, “A basic theorem of complementarity for the generalized variational-like inequality problem”. Jou. Mathem. Analysis and Appls., Vol. 158, 1991, pp. 124–138.
J. C. Yao, “Abstract variational inequality problems and a basic theorem of complementarity”. Computers Mathem. Appls., Vol. 25, 1993, pp. 73–79.
S. Y. Yu and J. C. Yao, “On vector variational inequalities”. Jou. Optimiz. Theory Appls., Vol. 89, 1996, pp. 749–769.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Kluwer Academic Publishers
About this chapter
Cite this chapter
Ansari, A.H., Siddiqi, A.H., Yao, JC. (2000). Generalized Vector Variational-Like Inequalities and their Scalarizations. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria. Nonconvex Optimization and Its Applications, vol 38. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0299-5_2
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0299-5_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7985-0
Online ISBN: 978-1-4613-0299-5
eBook Packages: Springer Book Archive